Perpendiculars: If we look around us, we will find countless examples of perpendicular lines. The corners of the wall, the English alphabet \(L\), the tiles of our kitchen, the hands of a clock when it hit exactly \(3\) o’clock, the corners of your study table are some examples. If two lines intersect each other at a right angle, then the lines are said to be perpendicular.

A line is said to be perpendicular to another line if two lines intersect at a right angle. Clearly, the first line is perpendicular to the second line if (1) the two lines meet, and (2) at the point of intersection a straight angle on one side of the first line is cut by the second line into two congruent angles. In this article, we will discuss perpendiculars, their properties, and perpendicular bisector. Read on to find more!

Perpendiculars Definition

Two lines (or rays or segments) are said to be perpendicular if they intersect such that the angles formed between them are right angles. In the figure, the lines \(l\) and \(m\) are perpendiculars. Symbolically it is written as \(l \bot m\).

Examples of Perpendiculars

We can give so many examples for perpendicular lines (or line segments) from things we come across in our daily life.

In this figure, the arm is perpendicular to the body. Therefore, the angle formed by the arm with the body is exactly \(90^\circ \) which is called a right angle.

The edges of a postcard meet at a right angle.

The English alphabet \(T\) is an example of perpendicular lines.

Perpendicular Through a Point on a Given Line

Draw a line \(l\) on the tracing paper. Mark a point \(P\) lying on this line. Now, we want to draw a perpendicular on \(l\) through \(P\). We fold the paper at point \(P\) such that the lines on both sides of the fold coincide with each other. When we unfold it, we find that the crease on the paper is perpendicular to the line \(l\) on it.

Drawing Perpendicular Using a Ruler and a Set-square

Steps of Construction:

Step 1: A line \(l\) is given, and a point \(P\) lies on it. Step 2: Place a ruler on the line \(l\) such that the point \(P\) is on the ruler. Hold this tightly.

Step 3: Now, place a set square along the ruler’s edge such that the right-angled corner of the set square is in contact with the ruler.

Step 4: Gently slide the set-square along the right direction to the edge of the ruler until its right-angled corner coincides with the point \(P\). Step 5: Hold the set-square tightly in this position. Draw a line \(PQ\) along the edge of the set square such that \(PQ\) is the perpendicular line to \(l\).

Perpendicular to a Line Through a Point Which is Not on It

Steps of construction:

Step 1: Draw a line \(l\), and a point \(A\) is outside the line \(l\). Step 2: Take \(A\) as a centre and draw an arc that cut the given line \(l\) at two distinct points \(M\) and \(N\). Step 3: Now use the same radius and with \(M\) and \(N\) as centres, draw two arcs that meet at a point, say \(B\) on the side opposite to the point \(A\) to the line \(l\). Step 4: Now join the points \(A\) and \(B\). The line \(AB\) is a perpendicular line of the given line \(l\).

Method of Ruler and Set Square

Suppose \(PQ\) is a line segment, and \(M\) is a point outside the line segment.
Steps of Construction
Step 1: Put the ruler on the paper so that it is lined up to the line segment \(PQ\).

Step 2: Keep one perpendicular side of the set square along the ruler. Be careful that the ruler doesn’t slide. The other side of the set square is now perpendicular to the ruler.

Step 3: Hold the ruler firmly on the paper and slide the set square gently along the ruler in such a way that the perpendicular side of the set square touches the point \(M\).
Step 4: Now, draw a line segment from the point \(M\) along the perpendicular side of the set square.
Step 5: This line segment \(ML\) is the required perpendicular to the line \(PQ\).

Properties of Perpendiculars

The properties of the perpendiculars are:
(i) Perpendicular lines always intersect at a right angle or \(90^\circ \).
(ii) If two lines are perpendicular to the same line, they are parallel.

Perpendicular Bisector

The perpendicular bisector of a line segment is perpendicular to the line segment that divides it into two equal parts.

Constructing Perpendicular Bisector of the Given Line Segment

Method of Ruler and Compass

Steps of Construction:

Step 1: First, draw a line segment \(AB\).
Step 2: Now set the compasses as radius more than half of the length of the line segment \(AB\).
Step 3: Take \(A\) as the centre, draw two arcs such that one arc is below and the other is above the line segment \(AB\).
Step 4: Take the same radius and \(B\) as centre draw two arcs above and below the line segment to cut the previous arcs at points \(M\) and \(N\) respectively.
Step 5: Finally, join the points \(M\) and \(N\). Then, the line \(l\) is the required perpendicular bisector of the line segment \(AB\). Line \(l\) intersects line segment \(AB\) at point \(P\).

Practice Exam Questions

Difference Between Perpendicular and Parallel Lines

The perpendicular lines always intersect at a right angle, and the parallel lines are always at the same distance from each other, and they do not intersect at any point. Therefore, the angle between two parallel lines is zero.

Use of Perpendiculars in Computing Distance

The distance from a point to a line is the distance to the nearest point on that line. It is the point from which a segment to a given point is perpendicular to the line. Similarly, the distance from a point to a curve is measured by a line segment that is perpendicular to the tangent line to the curve at the nearest point on the curve.

Vertical regression fits the data points by minimizing the sum of the squared perpendicular distances from the data points to the line. The distance from a point to a plane is measured as the length from a point along a segment that is perpendicular to the plane, meaning that it is perpendicular to all lines in the plane that go from the nearest point in the plane to a given point.

Solved Examples – Perpendiculars

Q.1. In the diagram given below, the line \(l\) is perpendicular to line \(m\).

For which two line segments, \(PE\) is the perpendicular bisector?
Ans: We know a line segment that divides a line into two equal parts is known as a perpendicular bisector.

Observing the figure, we can say,
Here, \(DE = EF {\text{}}\) and \(\overline {DF} \bot \overline {PE} \),
\(CE = EG {\text{}}\) and \(\overline {CG} \bot \overline {PE} \).
Hence, \(PE\) is the perpendicular bisector of the line segments \(\overline {DF} ,\,\overline {CG} \).

Q.2. There are two set-squares in your box. Find the measures of the angles that are formed at their corners? Write the measure of the common angle (if any).
Ans: Let us consider two set-squares given below.

The angles in the one set-square are \(45^\circ ,\,90^\circ ,\,45^\circ \) and in the other set-square are \(60^\circ ,\,90^\circ ,\,30^\circ \).
They have only \(90^\circ\) as a common angle.

Q.3. The line segment \(\overline {PQ} \) is perpendicular to the line segment \(\overline {XY} \). Let \(\overline {PQ} \) and \(\overline {XY} \) intersect at the point \(?\). Find the measure of \(\angle PAY\)
Ans: According to the question,
Let \(\overline {PQ} \) be the perpendicular to the line segment \(\overline {XY} \). Let \(\overline {PQ} \) and \(\overline {XY} \) intersect at point \(A\).

If a line segment is perpendicular to another line segment, then the angle between them will be a right angle.
Therefore, if two line segments intersect each other at \(90^\circ \), then the line segments are perpendicular to each other.
Hence, from the figure, it is clear that \(\angle PAY = 90^\circ \).

Q.4. Identify the pair of parallel and perpendicular line segments in the shape given below. Name them.

Ans: Perpendicular line segments: There are four pairs of perpendicular line segments.
\(\overline {HB} \bot \overline {BC} ,~\overline {BC} \bot \overline {CD} ,~\overline {FG} \bot \overline {GH} \) and \(\overline {GH} \bot \overline {HB} \)
Parallel line segments: There is only one pair of parallel line segments.
\(\overline {HG} \parallel \overline {BC} \)

Question 5: Determine whether the letter \(L\) model is perpendicular lines or not.
Ans: We know that parallel lines never meet, but perpendicular lines meet or intersect each other. The angle between two perpendicular lines always equals \(90^\circ \). Two line segments form the letter \(L\).

These two line segments are perpendicular to each other.
Hence, the line segments forming the letter \(L\) are perpendicular to each other.

Summary

Throughout this article, you learnt about perpendiculars, their definition, properties of perpendicular, and the construction of perpendicular line when a point is on the line (with the help of a ruler and a compass) and when the point is outside (with the help of a ruler and a set square). Also, we have discussed that the perpendicular bisector of a line segment divides the line segment into two equal parts, how to construct a perpendicular bisector of a line segment with the help of a compass, the difference between perpendicular and parallel lines, and some solved examples.

FAQs

Q.1. What is the perpendicular symbol?
Ans: The symbol for perpendicular is \( \bot \). It means perpendicular to.
For example: if a line \(l\) is perpendicular to a line \(m\) then we write symbolically \(l \bot m\).

Q.2. What is an example of a perpendicular line?
Ans: A good example of perpendicular lines we can see in the real world is the football field. All four corners of a football field are perpendicular to each other.

Q.3. Do parallel lines exist?
Ans: Yes, the parallel lines do exist in Euclidean geometry. A real-life example of parallel lines is railway tracks.

Q.4. Can parallel lines be perpendicular?
Ans: No. Parallel lines are always equidistant from each other.

Q.5. How do you know if a line is perpendicular to another line?
Ans: If two lines are perpendicular, then it means it is intersecting at a right angle.

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